In their reference model, seed BH formation is activated in metal-poor halos with metallicity \(Z<Z_{cr}=10^{-3.8} \, Z_\odot \) , independently of the seed mass. Then, the emergence of a specific seed flavour depends on the LW flux level and on the ISM dust-to-gas mass ratio (see [1]} and [2]} for details). Light seeds of \([40-140] M_\odot \)  and \(>260 M_\odot \)  form if both the dust-to-gas mass ratio of the ISM and the LW flux are relatively low, i.e. if \(\mathcal {D} \le 4.4 \times 10^{-9}\)  and \(J_{LW}< J_{cr}\) . More massive seeds are instead assumed to form in atomic cooling halos illuminated by a super-critical LW radiation (\(J_{LW}\ge J_{cr}\) ): medium-weight seeds of \(10^3 M_\odot \)  form in the presence of dust (\(\mathcal {D} \ge 4.4 \times 10^{-9}\) ) whereas \(10^5 M_\odot \)  heavy seeds form in dust-poor/free halos (\(\mathcal {D} < 4.4 \times 10^{-9}\) ). The results of this reference model, called R300 (to indicate the adopted LW critical threshold \(J_{cr}=300 \times 10^{-21}\rm erg/s/Hz/cm^2/sr\) ), are shown in the upper panels of Figure REF .
